Welch-Costas array

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A Welch-Costas array, or just Welch array, is a Costas array generated using the following method, first discovered by Lloyd R. Welch.

We take a primitive element, α, of a prime, p. We raise α to successive powers, modulo p. This creates a Costas permutation of length p-1. More formally, the dots defined by (i, α^i-1+c mod p) form a Welch array.

Example:

3 is a primitive element of 5.

3^1 = 3

3^2 = 9 = 4 (mod 5)

3^3 = 27 = 2 (mod 5)

3^4 = 81 = 1 (mod 5)

Therefore [3 4 2 1] is a Costas permutation. More specifically, this is an exponential Welch array. The transposition of the array is a logarithmic Welch array.

The number of Welch-Costas arrays which exist for a given size depends on the totient funtion.

[edit] References

• S. Golomb and H. Taylor, Constructions and Properties of Costas Arrays, PROC. IEEE, 72, 9, SEPTEMBER 1984